6.3 trigonometric functions (part 1)
TRANSCRIPT
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• Length of arc• Angles and measurements (radian
and degree)
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2 2
1 x y
1,0
2
3
2
6
3
4
Denote the unit circle as Uand the length of arc as t
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2 2
1 x y
1,0
3
2
2
6
3
4
Denote the unit circle as Uand the length of arc as t
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Find
12t
1,0
Initial point
Terminal point
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Find 2t
1,0
Initial point
Terminal point
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5
Find
8t
1,0Initial point
Terminal point
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Find 3t
1,0Initial point
Terminal point
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11Find
3t
1,0Initial point
Terminal point
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Angles and Measurements
Definition:
In geometry, angle is thoughtof as union of two rays calledthe sides, having a commonendpoint called the vertex.
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1,0
Initial side
Terminal side
Denote the angle as θ (or any Greek letters)
t
angle θ
Length of arc t
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1,0
Initial side Terminal side
t
angle θ Length of arc t
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Angles and Measurements
Definition:
If the terminal side lies onan axis, the angle is said tobe quadrantal .
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1,0
Initial side
Terminal side
t
quadrantal angle θ
Length of arc t
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Angles and Measurements
Definition:
The measurement of the anglefor which t=1 is called aradian .
It is written as Rm t
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1,0
1 radian
1t
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1,0
2t
2 Rm
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1,0
3.25t 3.25 Rm
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Angles and Measurements
Note that one completerevolution of the terminal sidefrom the initial side in thecounterclockwise direction is2 π .
More than one complete
revolution generates an angleof radian measure greater than2 π .
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1,0
1
4
9
4t
9
4
7
4
Angles having the same terminal side are called coterminal .
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1,0
23
23
t
Find the radian measure of the smallest positive angle that iscoterminal with the angle having the given radian measure.
2 423 3
Rm
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1,0
11
4
114
t
Find the radian measure of the smallest positive angle that iscoterminal with the angle having the given radian measure.
11 324 4
Rm
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1,0
0.54
0.54t
Find the radian measure of the smallest positive angle that iscoterminal with the angle having the given radian measure.
6.28 0.54 5.74 Rm
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Angles and Measurements
Definition:
Another unit of anglemeasurement is the degree .
11 where C is the circumference of a circle
360C
180 radian 1 radian
180
1801 radian
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1,0
1 radian
1t 180
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1 radian180
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Angles and Measurements
Corresponding degree and radianmeasures for certain angles.
Degreemeasure
30 45 60 90 120 135 150 180 270 360
Radianmeasure 16
14
13
12
23
34
56
32
2
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Angles and Measurements
We use the notation toindicate degree measure ofangle θ . It then follows that:
m
180 Rm m
180
Rm m
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Angles and Measurements
Example:
Find the degree measure to thenearest hundredths of a degreefor the angle having the givenradian measure (let π =3.1416).
57
Rm
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Angles and Measurements
Solution: Thus, the anglemeasure of
is
5
7
Rm
180
180 5
7
9007
128.57
R
m m
128.57
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Angles and Measurements
Example:
Find the degree measure to thenearest hundredths of a degreefor the angle having the givenradian measure (let π =3.1416).
0.3826 Rm
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Angles and Measurements
Solution: Thus, the anglemeasure of
is 0.3826 Rm
180
1800.3826
21.92
Rm m
21.92
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Angles and Measurements
We may transform degree anglemeasure which has decimal usingminutes and seconds. That is,
11'
60
1 11'' '
60 3600
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Angles and Measurements
Example:
14 4626 14 '46" 26 60 3600
26 0.233 0.013
26.25
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Assignment (1 whole)
Answer Exercises 6.1 #s 1,3,5,7,15, 17, 19, 21, 23, 25 (page298-299).